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## 8-1

**Additional Vocabulary Support **

_{Inverse Variation}

**Choose the expression from the list that best matches each sentence.**

** combined ** **constant of ** **inverse ** **joint **
** variation ** **variation ** **variation ** **variation**

** 1. **equations of the form *xy* 5*k*

** 2. **when one quantity varies with respect to two or more quantities

** 3. **when one quantity varies directly with two or more quantities

** 4. **the product of two variables in an inverse variation

**Choose the expression from the list that best matches each sentence.**

** combined ** **constant of ** **inverse ** **joint **
** variation ** **variation ** **variation ** **variation**

** 5. **When one quantity increases and the other quantity decreases proportionally,
** ** the relationship is an .

** 6. **Th e function *z* 5*kxy*is an example of a .

** 7. **Th e is represented by the variable *k*.

** 8. **Both functions *z* 5* _{wy}kx* and

*z*5

*kxy*are examples of .

**Multiple Choice**

** 9. **Which function would be used to model the relationship “*x* and *y* vary
inversely”?

*y* 5*kx* *z* 5*kxy* *y* 5*k _{x}*

*y*5

*x*

*k*

** 10. **Which function would be used to model the relationship “*z* varies jointly with

*x* and *y*”?

*y* 5*kxz* *z* 5*kxy* *y* 5_{xz}k*y*5 *xz*

*k*

**inverse variation**

**joint variation**

**constant of variation**

**combined variation**
**inverse variation**

**combined variation**

**joint variation**

**constant of variation**

**C**

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Th e spreadsheet shows data that could be modeled by an

equation of the form *PV*5*k*. Estimate *P* when *V*5 62.

**Understanding the Problem**

** 1. **Th e data can be modeled by

### z

_{ }

### z

*.*

** 2. **What is the problem asking you to determine?

**Planning the Solution**

** 3. **What does it mean that the data can be modeled by an inverse variation?

**.**

** 4. **How can you estimate the constant of the inverse variation?

.

** 5. **What is the constant of the inverse variation?

** 6. **Write an equation that you can use to fi nd *P* when *V*562.

**Getting an Answer**

** 7. **Solve your equation.

** 8. **What is an estimate for *P* when *V*562?

## 8-1

**Think About a Plan **

_{Inverse Variation}

**A** **B**

140.00 100 2

3 147.30 95 4 155.60 90

164.70 85 5

175.00 80 6

186.70 75 7

1 P V

* PV*5

**k****an estimate of the value of P when V**5**62**

**about 14,000**

**about 226**

**62P**5**14,000**
**The product of each pair of P and V values is approximately the same constant**

**Find PV for each row of the data. It should be approximately the same for **

**each row**

**62P**5**14,000**

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## 8-1

**Practice **

*Form G*

### Inverse Variation

**Is the relationship between the values in each table a ****direct variation****, an ****inverse ****variation****, or ****neither****? Write equations to model the direct and inverse variations.**

** 1. ** **x**

* y*
2
10

4 5

5 4

20 1

**2. ** **x**

* y*
1
2

3 8

7 20

10 29

** 3. ** **x**

* y*
1

6 2

12 5

30 7

42

**4. ** **x**

* y*
0.2

25 0.5

62.5 2

250 3

375

** 5. **
**x**

* y* 31 7 3
2

1 10

1 2

3 2

5 2

**6. ** **x**

* y*
3
5

1.5 10

0.5 30

0.3 50

**Suppose that ****x**** and ****y**** vary inversely. Write a function that models each inverse **
**variation. Graph the function and fi nd ****y**** when *** x*5

**10.**

** 7. ***x*57 when *y* 52 **8. ***x*54 when *y* 50.2 **9. ***x*51_{3} when *y*5 _{10}9

** 10. **Th e students in a school club decide to raise money by selling hats with the
school mascot on them. Th e table below shows how many hats they can
expect to sell based on how much they charge per hat in dollars.

**Price per Hat (p) ** 5 6 8 9

72 60 45 40
**Hats Sold (h) **

** a. **What is a function that models the data?

** b. **How many hats can the students expect to sell if they charge $7.50 per hat?

** 11. **Th e minimum number of carpet rolls *n* needed to carpet a house varies
directly as the house’s square footage *h* and inversely with the square footage

*r* in one roll. It takes a minimum of two 1200-ft2 carpet rolls to cover 2300 ft2 of
fl oor. What is the minimum number of 1200-ft2 carpet rolls you would need to
cover 2500 ft2 of fl oor? Round your answer up to the nearest half roll.

* y*5

**14**

_{x}**; 7**

_{5}*5*

**y**

_{5x}4; 0.08 or_{25}2*5*

**y**

_{10x}3**; 0.03 or**

_{100}3* ph*5

**360 or h**5

**360**

_{p}**48**

**2.5**
**inverse; y**5**20**_{x}**neither**

**direct; y**5**125x**

**inverse; *** y*5

**15**

_{x}**direct; y**5

**6x**

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## 8-1

**Practice **

(continued) *Form G*

### Inverse Variation

** 12. **On Earth, the mass *m *of an object varies directly with the object’s potential
energy *E* and inversely with its height above the Earth’s surface *h*. What is an
equation for the mass of an object on Earth? (*Hint:E*5*gmh*, where *g* is the
acceleration due to gravity.)

**Each ordered pair is from an inverse variation. Find the constant of variation.**

** 13. ** Q3, 1_{3}R **14. **(0.2, 6) **15. **(10, 5) **16. **Q5_{7}, 2_{5}R

** 17. **(213, 22) **18. **Q1_{2}, 10R **19. **Q1_{3}, 6_{7}R

** 20. **(4.8, 2.9) **21. **_{Q}5_{8},22_{5}_{R} **22. (4.75, 4)**

**Write the function that models each variation. Find ****z****when *** x*5

**6 and**

*5*

**y****4.**

** 23. ***z* varies jointly with *x* and *y*. When *x*5 7 and *y*5 2, *z*5 28.

** 24. ***z* varies directly with *x* and inversely with the cube of *y*. When *x*5 8 and *y*5 2,

*z*5 3.

**Each pair of values is from an inverse variation. Find the missing value.**

** 25. **(2, 4), (6, *y*) **26. **Q1_{3}, 6R, Q*x*, 21_{2}R **27. (1.2, 4.5), (2.7, ***y*)

** 28. **One load of gravel contains 240 ft3 of gravel. Th e area *A* that the gravel will
cover is inversely proportional to the depth *d* to which the gravel is spread.
** a. **Write a model for the relationship between the area and depth for one load

of gravel.

** b. **A designer plans a playground with gravel 6 in. deep over the entire play
area. If the play area is a rectangle 40 ft wide and 24 ft long, how many loads
of gravel will be needed?

* m*5

**kE**_{h}**, where k**5

**1**

_{g}2**286**

**1** **1.2** **50** **2 _{7}**

**13.92**

**5**

2**1 _{4}**

**2**
**7**

**19**

2**4** **2**

**2 loads**
**4**

**3**

* z *5

**2xy; 48**

* z*5

**3x**

**y****3;**

**9**
**32**

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## 8-1

**Practice **

*Form K*

### Inverse Variation

**Is the relationship between the values in each table a ****direct variation****, an **
**inverse variation****, or ****neither****? Write an equation to model the direct and inverse **
**variations.**

** 1. ** _{x}_{y}

0.1 3 6 24

3 0.1 0.05 0.0125

**2. ** _{x}_{y}

1 2 5 6

3 6 15 18

**3. ** _{x}_{y}

0 2 4 6

1 5 7 8

**Suppose that ****x**** and ****y**** vary inversely. Write a function that models each inverse **
**variation. Graph the function and fi nd ****y**** when *** x *5

**10.**

** 4. ***x*52 when *y* 5 24 **5. ***x*5 29 when *y* 5 21 **6. ***x*51.5 when *y* 510

** 7. **Suppose the table at the right shows the
time *t* it takes to drive home when you travel
at various average speeds *s*.

** a. **Write a function that models the relationship
between the speed and the time it takes to
drive home.

** b. **At what speed would you need to drive to
get home in 50 min or 5_{6} h?

**Time ****t ****(h)** **Speed ****s ****(mi/h)**
60

40

30

13.3

1 6 1 4 1 3 3 4

*O*

*x*
*y*
4

4 8

4 8

4 *O* *x*

*y*

4

4

4 4 *O* *x*

*y*

4

4
4 4
**inverse variation; *** y*5

**0.3**

_{x}* y*5 2

**8**

_{x}**;**2

**4**

_{5}**direct variation; y**5**3x**

* y*5

**9**

_{x}**;**

_{10}9* s*5

**10**

_{t}**12 mi/h**

**neither**

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**Use combined variation to solve each problem.**

** 8. **Th e height *h* of a cylinder varies directly with the volume of the cylinder and
inversely with the square of the cylinder’s radius *r *with the constant equal to 1_{p}.
** a. **Write a formula that models this combined variation.

** b. **What is the height of a cylinder with radius 4 m and volume 500 m3?
Use 3.14 for p and round to the nearest tenth of a meter.

** 9. **Some students volunteered to clean up a highway near their school. Th e
amount of time it will take varies directly with the length of the section
of highway and inversely with the number of students who will help. If
25 students clean up 5 mi of highway, the project will take 2 h. How long
would it take 85 students to clean up 34 mi of highway?

**Write the function that models each variation. Find ****z**** when *** x*5

**2 and**

*5*

**y****6.**

** 10. ***z* varies inversely with *x* and directly with *y*. When *x* 55 and *y*5 10, *z*5 2.

** 11. ***z* varies directly with the square of *x* and inversely with *y*. When *x*52 and

*y*5 4, *z* 53.

**Each ordered pair is from an inverse variation. Find the constant of variation.**

** 12. ** (2, 2) **13. **(1, 8) **14. **(9, 4)

**Each pair of values is from an inverse variation. Find the missing value.**

** 15. ** (9, 5), (*x*, 3) **16. **(8, 7), (5, *y*) **17. **(2, 7), (*x*, 1)

## 8-1

**Practice **

(continued) *Form K*

### Inverse Variation

* h*5

**V**π**r****2**

* z*5

**y**_{x}**; 3**

* z*5

**3x**

_{y}**2; 2**

* k*5

**4**

* x*5

**15**

* k*5

**8**

* y*5

**11.2**

* k*5

**36**

* x*5

**14**

**10.0 m**

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## 8-1

**Standardized Test Prep**

_{Inverse Variation}

**Multiple Choice**

**For Exercises 1−5, choose the correct letter.**

** 1. **Which equation represents inverse variation between *x* and *y*?

4*y*5 *kx* *xy* 54*k* *y* 54*kx* 4*k* 5*x _{y}*

** 2. **Th e ordered pair (3.5, 1.2) is from an inverse variation. What is the constant of
variation?

2.3 2.9 4.2 4.7

** 3. **Suppose *x* and *y* vary inversely, and *x*5 4when *y* 59. Which function
models the inverse variation?

*y* 536_{x}*x* 5_{36}*y* *y* 5_{36}*x* *x _{y}* 536

** 4. **Suppose *x* and *y* vary inversely, and *x*5 23 when*y* 51_{3}. What is the value of

*y* when *x* 59?

29 21 21_{9} 1_{9}

** 5. **In which function does *t* vary jointly with *q* and *r* and inversely with *s*?

*t* 5*kq _{rs}*

*t*5

_{qr}ks*t*5

*s*

*kqr* *t*5

*kqr*
*s*

**Short Response**

** 6. **A student suggests that the graph at the right represents the
inverse variation *y* 53* _{x}*. Is the student correct? Explain.

2 2 4 6

4 6 8

*y*

*x*

**B**

**H**

**A**

**H**

**D**

**No; for each point on the graph xy**5**6, not 3.**
**[2] correct answer with explanation**

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## 8-1

**Enrichment**

_{Inverse Variation}

**Each situation below can be modeled by a direct variation, inverse variation, **
**joint variation, or combined variation equation. Decide which model to use and **
**explain why.**

** 1. **Th e circumference *C* of a circle is about 3.14 times the diameter *d*.

** 2. **Th e number of cavities that develop in a patient’s teeth depends on the total
number of minutes spent brushing.

** 3. **Th e time it takes to build a bridge depends on the number of workers.

** 4. **Th e number of minutes it will take to solve a problem set depends on the
number of problems and the number of people working on the problem set.

** 5. **Th e current *I* in an electrical circuit decreases as the resistance *R* increases.

** 6. **Charles’s Gas Law states the volume *V* of an enclosed gas at a constant
pressure will increase as the absolute temperature *T* increases.

** 7. **Boyle’s Law states that the volume *V *of an enclosed gas at a constant
temperature is related to the pressure. Th e pressure of 3.45 L of neon gas is
0.926 atmosphere (atm). At the same temperature, the pressure of 2.2 L of
neon gas is 1.452 atm.

**Direct variation; answers may vary. Sample: This relationship can be modeled **
**by the equation C **5** 3.14d, where 3.14 is the constant of variation.**

**Inverse variation; answers may vary. Sample: As the number of minutes spent **
**brushing increases, the number of cavities should decrease. This suggests an **
**inverse variation.**

**Inverse variation; answers may vary. Sample: As the number of workers **
**increases, the time it takes to build a bridge should decrease. This suggests an **
**inverse variation.**

**Combined variation; answers may vary. Sample: The time to solve a problem **
**set increases as the number of problems increases, but decreases as the number **
**of people working on the set increases. This suggests a combined variation.**

**Inverse variation; answers may vary. Sample: As one variable increases, the **
**other decreases. This suggests an inverse variation.**

**Direct variation; answers may vary. Sample: As one variable increases, so does **
**the other. This suggests a direct variation.**

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## 8-1

**Reteaching **

_{Inverse Variation}

Th e fl owchart below shows how to decide whether a relationship between two variables is a direct variation, inverse variation, or neither.

How does the value of *x* change?

How does the value of *y* change? How does the value of *y* change?

yes _{yes}

no no

increases decreases

increases decreases increases

neither direct variation inverse variation neither decreases

Test *x* and *y* values in direct
*y*
*x*
variation model: _{k }_{ }_{.}

Test * _{x}* and

*values in inverse variation model:*

_{y}*xy*

*k.*

Is each ratio of * _{y}* to

*equal to the same value,*

_{x}*?*

_{k}Is each product of * _{x}* and

*equal to the same value,*

_{y}*?*

_{k}** Problem**

Do the data in the table represent a direct variation, inverse variation, or neither?

**x****y**

1

20 2

10 4

5 5

4

As the value of *x* increases, the value of *y* decreases, so test the table values in
the inverse variation model: *xy* 5*k* : 1 ? 20520, 2? 10520, 4 ? 55 20,
5 ? 4 520. Each product equals the same value, 20, so the data in the table model
an inverse variation.

**Exercises**

**Do the data in the table represent a direct variation, inverse variation, or neither?**

** 1. ** **x**

* y*
5

10 10

20 15

30 20

40 **2. **

**x****y**

1

12 3

4 4

3 6

2

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## 8-1

**Reteaching **

(continued)
### Inverse Variation

To solve problems involving inverse variation, you need to solve for the constant
of variation *k* before you can fi nd an answer.

** Problem**

Th e time *t* that is necessary to complete a task varies inversely as the number of
people *p* working. If it takes 4 h for 12 people to paint the exterior of a house, how
long does it take for 3 people to do the same job?

*t* 5* _{p}k* Write an inverse variation. Because time is dependent on people, t is the dependent variable
and p is the independent variable.

4 5_{12}*k* Substitute 4 for t and 12 for p.

485*k* Multiply both sides by 12 to solve for k, the constant of variation.

*t* 548* _{p}* Substitute 48 for k. This is the equation of the inverse variation.

*t* 548_{3} 516 Substitute 3 for p. Simplify to solve the equation.

It takes 3 people 16 h to paint the exterior of the house.

**Exercises**

** 3. **Th e time *t* needed to complete a task varies inversely as the number of people

*p*. It takes 5 h for seven men to install a new roof. How long does it take ten
men to complete the job?

** 4. **Th e time *t* needed to drive a certain distance varies inversely as the speed *r*. It
takes 7.5 h at 40 mi/h to drive a certain distance. How long does it take to drive
the same distance at 60 mi/h?

** 5. **Th e cost of each item bought is inversely proportional to the number of items
when spending a fi xed amount. When 42 items are bought, each costs $1.46.
Find the number of items when each costs $2.16.

** 6. **Th e length *l* of a rectangle of a certain area varies inversely as the width *w*. Th e
length of a rectangle is 9 cm when the width is 6 cm. Determine the length if
the width is 8 cm.

**3.5 h**

**5 h**

**about 28 items**

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## 8-2

**Additional Vocabulary Support **

_{The Reciprocal Function Family}

**For Exercises 1–5, draw a line from each word in Column A to its defi nition in **
**Column B. **

**Column A ** ** Column B**

** 1. **reciprocal **A. **function that models inverse variation

** 2. **branch **B. ** stretches, compressions, refl ections, and
horizontal and vertical translations

** 3. **reciprocal function **C. **multiplicative inverse

** 4. **refl ection of the reciprocal function **D. ** the graph of *y* 5 21_{x}

** 5. **transformations **E. ** each part of the graph of a reciprocal
function

**For Exercises 6–9, the graph of each function is a transformation of the parent **

**graph of ****f****(****x****)**5**1**_{x}**. Draw a line from each function to its transformation.**

** 6. ***f*(*x*) 52_{x}**A. **a horizontal translation

** 7. ***f*(*x*) 5 21_{x}**B. **a refl ection over the *x*-axis
** 8. ***f*(*x*) 5_{x}_{2}1 _{2} **C. **a vertical translation

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** a. Gasoline Mileage** Suppose you drive an average of 10,000 miles each year.

Your gasoline mileage (mi/gal) varies inversely with the number of gallons of gasoline you use each year. Write and graph a model for your average mileage

*m* in terms of the gallons *g* of gasoline used.

** b. **After you begin driving on the highway more often, you use 50 gal less per
year. Write and graph a new model to include this information.

** c. **Calculate your old and new mileage assuming that you originally used 400 gal
of gasoline per year.

** 1. **Write a formula for gasoline mileage in words.

**.**

** 2. **Write and graph an equation to model your average mileage *m*

in terms of the gallons *g* of gasoline used.

** **

** 3. **Write and graph an equation to model your average mileage *m* in
terms of the gallons *g* of gasoline used if you use 50 gal less per year.

** 4. **How can you fi nd your old and your new mileage from your
equations?

**.**

**5.** What is your old mileage?

**6. **What is your new mileage?

## 8-2

**Think About a Plan **

_{The Reciprocal Function Family}

0 100 200 300 400 0

10 20 30 40

Gas Used (gal)

Mileage (mi/gal)

*m*

*g*

0 100 200 300 400 0

10 20 30 40

*m*

*g*

Gas Used (gal)

Mileage (mi/gal)

**The mileage is equal to the number of miles divided by the number of gallons**

**Evaluate each equation at g**5**400**

**25 mi/gal**

**about 28.6 mi/gal**
* m*5

**10,000**

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## 8-2

**Practice **

*Form G*

### The Reciprocal Function Family

**Graph each function. Identify the ****x****- and ****y****-intercepts and the asymptotes of the **
**graph. Also, state the domain and the range of the function.**

** 1. ***y*5 12_{x}**2. ***y*5 5_{x}**3. ***y*5 24_{x}

**Use a graphing calculator to graph the equations *** y*5

**1**

_{x}**and**

*5*

**y**

**a**_{x}**using the**

**given value of**

**a****. Th en identify the eff ect of**

**a****on the graph.**

** 4. ***a*53 **5. ***a*5 25 **6. ***a*50.4

**Sketch the asymptotes and the graph of each function. Identify the domain and range.**

** 7. ***y*5 1* _{x}* 13

**8.**

*y*5

_{4}3

*11*

_{x}_{2}

**9.**

*y*5

_{x}_{2}3

_{1}12

**Write an equation for the translation of *** y*5 2

**3**

_{x}**that has the given asymptotes.**

**10.**

*x*5 21;

*y*53

**11.**

*x*54;

*y*5 22

**12.**

*x*50;

*y*5 6

6 3 6 9 9

3 6 9

*x*
*y*

*O*

*x* scale: 1* _{y}* scale: 1

*scale: 1*

_{x}*scale: 1*

_{y}4 4

*x*
*y*

*O*

*x* scale: 0.1*y* scale: 0.1
**no intercepts; x**5**0,**

* y*5

**0; all real numbers**

**except x**5

**0; all real**

**num. except y**5

**0**

**Stretch by a factor of 3.**

**all real numbers except **
* x*5

**0; all real numbers**

**except y**5

**3**

**all real numbers except **
* x*5

**0; all real numbers**

**except y**5

**1**

_{2}**all real numbers except **
* x*5

**1; all real numbers**

**except y**5

**2**

**no intercepts; x**5**0,**
* y*5

**0; all real numbers**

**except x**5

**0; all real**

**num. except y**5

**0**

**Reﬂ ect over x-axis and **
**stretch by a factor of 5.**

**no intercepts; x**5**0,**
* y*5

**0; all real numbers**

**except x**5

**0; all real**

**num. except y**5

**0**

**Shrink by a factor of **
**0.4.**

* y*5 2

_{x}_{1}

**3**

**1**

_{1}**3**

*5 2*

**y**

_{x}_{2}

**3**

**2**

_{4}**2**

*5 2*

**y****3**

*1*

_{x}**6**

*x*

*y*

*O*

1 2 2

2 3 4

*x*
*y*

*O* 4
4

4 4 6

*x*
*y*

*O* 1 2
1
2

2

3

*O*
2

2 2

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## 8-2

**Practice **

(continued) *Form G*

### The Reciprocal Function Family

** 13. **Th e length of a pipe in a panpipe / (in feet) is inversely proportional to its pitch
** ** *p* (in hertz). Th e inverse variation is modeled by the equation *p* 5495_{/} . Find

the length required to produce a pitch of 220 Hz.

**Write each equation in the form *** y*5

**k**_{x}**.**

** 14. ** *y*5 _{5}4_{x}**15. ***y*5 2_{2}7_{x}**16. ***xy* 5 20.03

**Sketch the graph of each function. **

** 17. ** *xy* 56 **18. ***xy* 11050 **19. **4*xy* 5 21

** 20. **Th e junior class is buying keepsakes for Class Night. Th e price of each

keepsake *p* is inversely proportional to the number of keepsakes *s* bought. Th e
keepsake company also off ers 10 free keepsakes in addition to the class’s
** ** order. Th e equation *p* 5* _{s}*1800

_{1}

_{10}models this inverse variation.

** a. **If the class buys 240 keepsakes, what is the price for each one?

** b. **If the class pays $5.55 for each keepsake, how many can they get, including
the free keepsakes?

** c. **If the class buys 400 keepsakes, what is the price for each one?
** d. ** If the class buys 50 keepsakes, what is the price for each one?

**Graph each pair of functions. Find the approximate point(s) of intersection. **

** 21. ** *y*5 _{x}_{2}3 _{4}; *y* 52 **22. ***y*5 _{x}_{1}2 _{5}; *y* 5 21.5
2 6

*O*
2
6

*x*
*y*

2
*O*

2 4

4 8

*x*
*y*

0.5
*O*
0.5
1

1

*x*
*y*
**2.25 ft**

**$7.20**

**324**

**$4.39**

**(5.5, 2)** **(**2**6.3, **2**1.5)**
**$30**

* y*5

**0.8**

_{x}*52*

**y****3.5**

_{x}*52*

**y****0.03**

_{x}Intersection
*x*5 5.5 *y*5 2
*x *scale: 1 *y *scale: 1

Intersection

**Prentice Hall Foundations Algebra 2 • **Teaching Resources
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## 8-2

**Practice **

*Form K*

### The Reciprocal Function Family

**Graph each function. Identify the ****x**- **and ****y****-intercepts and asymptotes of the **
**graph. Also, state the domain and range of the function. **

** 1. ***y*5 22_{x}**2. ***y*5 4_{x}**3. ***y*5 25_{x}

**Graph the equations *** y*5

**1**

_{x}**and**

*5*

**y**

**a**_{x}**using the given value of**

**a****. Th en identify**

**the eff ect of**

**a****on the graph.**

** 4. ***a*5 23 **5. ***a*54 **6. ***a*5 20.25

**Sketch the asymptotes and the graph of each function. Identify the domain **
**and range.**

** 7. ***y*5 1* _{x}* 12

**8.**

*y*5

_{x}_{2}1

_{2}13

**9.**

*y*5

*2*

_{x}_{1}10

_{1}28

*O* *x*
*y*
4
8
4
8
4
8

4 8 *O* *x*

*y*
4
4
8
8
4 *O*
*x*
*y*
8
4
8
4
8
8
**x-intercept: none; **

**y-intercept: none; horizontal ****asymptote: x-axis; vertical **
**asymptote: y-axis; domain: **
**all real numbers except **
* x*5

**0; range: all real**

**numbers except y**5

**0**

**reﬂ ected across the **
**x-axis and stretched ****by a factor of 3**

**domain: all real numbers **
**except 0; range: all real **
**numbers except 2**

**x-intercept: none; **

**y-intercept: none; horizontal ****asymptote: x-axis; vertical **
**asymptote: y-axis; domain: **
**all real numbers except **

* x*5

**0; range: all real**

**numbers except**

*5*

**y****0**

**stretched by a factor of 4**

**domain: all real numbers **
**except 2; range: all real **
**numbers except 3**

**x-intercept: none; **

**y-intercept: none; horizontal ****asymptote: x-axis; vertical **
**asymptote: y-axis; domain: **
**all real numbers except **
* x*5

**0; range: all real**

**numbers except y**5

**0**

**reﬂ ected across the **
**x-axis and compressed ****by a factor of 0.25**

**domain: all real numbers **
**except **2**1; range: all real **
**numbers except **2**8**

*O* *x*
*y*
4
8
4
8
4
8
4 8

* y*ⴝ

**1**

_{x}*ⴝⴚ*

**y****3**

_{x}*O* *x*
*y*
4
4
8
4
8
4

* y*ⴝ

**1**

_{x}*ⴝ*

**y****4**

_{x}*O* *x*
*y*
4
8
4
8
4
8
4 8

* y*ⴝ

**1**

_{x}*ⴝⴚ*

**y****0.25**

_{x}*O* *x*
*y*
4
4
8
4
8

4 *O* *x*

**Prentice Hall Foundations Algebra 2 • **Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
**Write an equation for the translation of *** y*5

**3**

_{x}**that has the given asymptotes.**

** 10. ** *x*50 and *y* 52 **11. ***x*5 22 and *y* 54 **12. ***x*55 and *y* 5 23

**Sketch the graph of each function.**

** 13. ** 3*xy* 51 **14. ***xy* 28 50 **15. **2*xy* 5 26

** 16. Writing** Explain the diff erence between what happens to the graph of
the parent function of *y* 5*a _{x}* when u

*a*u .1 and what happens when 0 , u

*a*u ,1.

** 17. ** Suppose your class wants to get your teacher an end-of-year gift of a weekend
package at her favorite spa. Th e package costs $250. Let *c* equal the cost each
student needs to pay and *s* equal the number of students.

** a. **If there are 22 students, how much will each student need to pay?
** b. **Using the information, how many total students (including those from

other classes) need to contribute to the teacher’s gift, if no student wants to pay more than $7?

** c. Reasoning** Did you need to round your answers up or down? Explain.

## 8-2

**Practice **

(continued) *Form K*

### The Reciprocal Function Family

* y*5

**3**

*1*

_{x}**2**

**When **»* a*… S

**1, the parent function is stretched by the factor of a. When 0**R »

*… R*

**a****1,**

**the parent function is compressed by the factor of a.**

**up; because if you round down, the total contribution by the students would be **
**less than $250.**

* y*5

_{x}_{1}

**3**

**1**

_{2}**4**

*5*

**y**

_{x}_{2}

**3**

**2**

_{5}**3**

*O*

*x*
*y*
4
8

4 8

4 8

4 8 *O* *x*

*y*

4

4

4 4 *O*

*x*
*y*

4 8

4 8

4 8

4 8

**$11.37**

**Prentice Hall Algebra 2 • **Teaching Resources
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## 8-2

**Standardized Test Prep**

_{The Reciprocal Function Family}

**Multiple Choice**

**For Exercises 1−3, choose the correct letter.**

** 1. **What is an equation for the translation of *y* 5 24.5* _{x}* that has asymptotes at

*x*53 and *y* 5 25?

*y* 5 2* _{x}*4.5

_{2}

_{3}25

*y*5 2

*4.5*

_{x}_{2}

_{5}13

*y* 5 2* _{x}*4.5

_{1}

_{3}25

*y*5 2

*4.5*

_{x}_{1}

_{5}13

** 2. **What is the equation of the vertical asymptote of *y* 5 _{x}_{2}2 _{5}?

*x* 5 25 *x* 50 *x*5 2 *x*55

** 3. **Which is the graph of *y* 5_{x}_{1}1 _{1}2 2?

**Extended Response**

** 4. **A race pilot’s average rate of speed over a 720-mi course is inversely
proportional to the time in minutes *t* the pilot takes to fl y a complete race
course. Th e pilot’s fi nal score *s* is the average speed minus any penalty points

*p* earned.

** a. **Write a function to model the pilot’s score for a given *t* and *p*. (*Hint:d*5 *rt*)
** b. **Graph the function for a pilot who has 2 penalty points.

** c. **What is the maximum time a pilot with 2 penalty points can take to fi nish
the course and still earn a score of at least 3?

*x*
*y*
*O*

1 2 1

1

2

4

3

5

2 2

4

2

4

*x*
*y*

*O*

2

3 1 1

3

4

5

*x*
*y*
*O*
*O*

2

4

4
1 *y* _{x}

**A**

**I**

**D**

**[4] s**5**720***_{t}* 2

**p;****[3] correct answer with most of work shown**

**and appropriate strategies used OR**

**incorrect answer with all work shown and**

**appropriate strategies used**

**[2] correct answer with little work shown OR **
**incorrect answer but work shown reﬂ ects **
**some understanding of problem**

**[1] answer is incomplete or incorrect and no **
**work is shown**

**[0] no answer given**
**144 min**

100 200 300 2

4
*s*

*O*

**Prentice Hall Algebra 2 • **Teaching Resources
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## 8-2

**Enrichment**

_{The Reciprocal Function Family}

**Understanding Horizontal Asymptotes**

Th e line *y* 5 3_{4} is a horizontal asymptote for the graph of the

function *y* 53_{4}*x _{x}*1

_{2}5

_{8}. By using long division, you can rewrite

this function in the form quotient 1 remainder divided by

the divisor: *y* 5 3_{4}1_{4}* _{x}*11

_{2}

_{8}.

Examine what happens to the remainder divided by the
divisor and the value of *y* as the value of *x* gets larger. Fill in
the following table to four decimal places.

Note that as *x* gets larger, both the remainder and the value of *y* get smaller.

Although the value of *y* is always greater than 3_{4}, it gets closer to 3_{4} as *x* gets

larger. As *x* gets infi nitely large, *y* approaches 3_{4} from above. Write this as:

As *x* S 1`, *y*S3_{4} from above.

Examine what happens as *x* gets smaller. Fill in the following table to four
decimal places.

Here the value of *y* is always less than 3_{4}, but it gets closer to 3_{4} as *x* gets smaller
(more negative). Write this as: As *x* S2`, *y*S3_{4} from below.

In both cases, *y* approaches 3_{4}, so the horizontal asymptote is *y* 53_{4}.

*O*

4 6

2

2

2

4

4 4 *x*

*y*

**1.**
**2.**
**3.**

**x**

3 10 100

11
4*x 2 *8

3 4

11
4*x 2 *8

*y*5 1

**4.**
**5.**
**6.**

**x**

23

210

2100

11
4*x 2 *8

3 4

11
4*x 2 *8

*y*5 1

**2.7500** **3.5000**

**0.3438** **1.0938**

**0.0281** **0.7781**

2**0.5500** **0.2000**

2_{0.2292}_{0.5208}

**Prentice Hall Algebra 2 • **Teaching Resources
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## 8-2

**Reteaching **

_{The Reciprocal Function Family}

**A Reciprocal Function in **
**General Form**

**Two Members of the **
**Reciprocal Function Family**

Th e*general form* is *y* 5 *a*

*x* 2*h* 1*k*, where

*a*20 and *x*2*h*.

When *a*21, *h* 50, and *k* 50, you get
the *inverse variation function,y* 5*a _{x}*.

Th e graph of this equation has a horizontal
asymptote at *y* 5*k* and a vertical asymptote
at *x*5 *h*.

When *a* 51, *h* 50, and *k* 50, you get
the *parent reciprocal function,y* 51* _{x}*.

** Problem**

What is the graph of the inverse variation function *y* 52* _{x}*5?

**Step 1** Rewrite in general form and identify *a*, *h*, and *k*.

*y* 5* _{x}*2

_{2}5

_{0}10

*a*5 25,

*h*50,

*k*5 0

**Step 2** Identify and graph the horizontal horizontal asymptote: *y* 5*k*

and vertical asymptotes. *y* 50

vertical asymptote: *x* 5*h*

*x* 50

**Step 3** Make a table of values for *y* 52* _{x}*5. Plot the points and then
connect the points in each quadrant to make a curve.

**y**

* x* 25 22.5 21 1 5

1 2 21 2.5

22

5 25

**Exercises**

**Graph each function. Include the asymptotes. **

** 1. ***y*5 9_{x}**2. ***y*5 24_{x}**3. ***xy* 52

5 4

*y*

4

4 3

3 3 2

2 2 1

1

5 3 1

*x*

*x*
*y*

*O*

4

6 2 6

2 6

6 4

9

3

9

3 6

6 9

9
*y*

*O*

*x*

2

1

2

1 2

2

*y*

**Prentice Hall Algebra 2 • **Teaching Resources
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A reciprocal function in the form *y* 5 _{x}*a*

2*h* 1 *k* is a *translation* of the inverse

variation function *y*5 *a _{x}*. Th e translation is

*h*units horizontally and

*k*units vertically. Th e translated graph has asymptotes at

*x*5

*h*and

*y*5

*k*.

** Problem**

What is the graph of the reciprocal function *y*5 2_{x}_{1}6 _{3} 12?

**Step 1** Rewrite in general form and identify *a*, *h*, and *k*.

*y* 5 26

*x* 2(23)1 2 *a*5 26, *h* 5 23, *k* 52

**Step 2** Identify and graph the horizontal horizontal asymptote: *y*5 *k *

and vertical asymptotes. *y*5 2

vertical asymptote: *x*5 *h*

*x*5 23

**Step 3** Make a table of values for *y* 52* _{x}*6, then

*translate*each (

*x*,

*y*) pair to (

*x*1

*h*,

*y*1

*k*). Plot the translated points and connect the points in each quadrant to make a curve.

* x *1

**2**

_{(}

_{3)}* y *1

**2**

**y*** x* 26

29

1

3

6

3

21

1

23

26

2

4

3

0

22

0

22

25

3

5 2

21

23

21

**Exercises**

**Graph each function. Include the asymptotes. **

** 4. ***y*5 _{x}_{2}3 _{2} 24 **5. ***y*5 2_{x}_{2}4 _{8} **6. ***y*5 _{3}2* _{x}* 13

_{2}8

8 *y*

4 6 6

6 4

4 2

8 _{}6 _{}2

*x*

## 8-2

**Reteaching **

(continued)
### The Reciprocal Function Family

*O*
4

6 8 10

4 8*x*

*y*

*O*
*y*

2 4 6

4

*x*

2 4 6 8

2 4

2 2

2
*x*
*y*

**Prentice Hall Algebra 2 • **Teaching Resources
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## 8-3

**Additional Vocabulary Support **

_{Rational Functions and Their Graphs}

**Concept List**

**Choose the concept from the list above that best represents the item in each box.**

** 1. **the line that a graph
approaches as *y*

increases in absolute value

** 2. **In the denominator,
these reveal the
points of discontinuity.

** 3. **Th is type of

discontinuity appears as a hole in the graph.

** 4. **Th is type of graph has
no jumps, breaks, or
holes.

** 5. **a function that you
can write in the form

*f*(*x*)5 *P*(*x*)

*Q*(*x*) where
** ** *P*(*x*) and *Q*(*x*) are

polynomial functions

** 6. **a graph that has a
one-point hole or a vertical
asymptote

** 7. **Th is type of

discontinuity appears as a vertical asymptote on the graph.

** 8. **Th e graph of *f *(*x*) is
not continuous at
this point.

** 9. **the line that a graph
approaches as *x*

increases in absolute value

** continuous ** **discontinuous ** **factors**

** horizontal asymptote non-removable discontinuity ** **point of discontinuity**
** rational function ** **removable discontinuity ** **vertical asymptote**

**vertical asymptote**

**continuous**

**non-removable **
**discontinuity**

**factors**

**rational function**

**point of discontinuity**

**removable **
**discontinuity**

**discontinuous**

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**Grades **A student earns an 82% on her fi rst test. How many consecutive 100% test
scores does she need to bring her average up to 95%? Assume that each test has
equal impact on the average grade.

**Understanding the Problem**

** 1. **One test score is

### z

_{ }

### z

.** 2. **Th e average of all the test scores is

### z

_{ }

### z

.** 3. **What is the problem asking you to determine?

**Planning the Solution**

** 4. **Let *x* be the number of 100% test scores. Write an expression for the total
number of test scores.

** 5. **Write an expression for the sum of the test scores.

** 6. **How can you model the student’s average as a rational function?

**Getting an Answer**

** 7. **How can a graph help you answer this question?

.

** 8. **What does a fractional answer tell you? Explain.

.

** 9. **How many consecutive 100% test scores does the student need to bring her
average up to 95%?

## 8-3

**Think About a Plan **

_{Rational Functions and Their Graphs}

**82%**

**95%**

**the number of tests with scores of 100% the student needs to have an **

**average of 95%**

**Answers may vary. Sample: I can graph the rational function and the **

**function y **5** 95 at the same time. The intersection is the solution**

**I would have to round a fractional answer up to the nearest whole number, because the **

**number of test scores must be a whole number and the average must be 95% or greater**
* x*1

**1**

**3**
**100x**1 **82**

**Prentice Hall Gold Algebra 2 • **Teaching Resources
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## 8-3

**Practice **

*Form G*

### Rational Functions and Their Graphs

**Find the domain, points of discontinuity, and ****x****- and ****y****-intercepts of each **
**rational function. Determine whether the discontinuities are removable or **
**nonremovable.**

** 1. ***y*5 (*x* 2* _{x}*4)(

_{1}

*x*

_{3}13)

**2.**

*y*5 (

*x*2

*3)(*

_{x}_{2}

*x*

_{2}11)

** 3. ***y*5 _{x}_{1}2 _{1} **4. ***y*5 4*x*

*x*4116

**Find the vertical asymptotes and holes for the graph of each rational function.**

** 5. ***y*5 52*x*

*x*221 **6. ***y*5

*x*222

*x*12

** 7. ***y*5 *x*

*x*(*x* 21) **8. ***y*5

*x*13

*x*229

** 9. ***y*5 *x* 22

(*x* 12)(*x*22) **10. ***y*5

*x*224

*x*214

** 11. ** *y*5 *x _{x}*22

_{2}25

_{4}

**12.**

*y*5 (

*x*22)(2

*x*13) (5

*x*14)(

*x*23)

**Find the horizontal asymptote of the graph of each rational function.**

** 13. ** *y*5 _{x}_{2}2 _{6} **14. ***y* 5*x _{x}*

_{2}12

_{4}

**15.**

*y*5 2

*x*2 13

*x*2 26 **16. ***y* 5

3*x*212

*x*222

**Sketch the graph of each rational function.**

** 17. ** *y*5 _{x}_{2}3 _{2} **18. ***y* 5 3

(*x*22)(*x* 12) **19. ***y*5

*x*

*x*214 **20. ***y* 5

*x* 12

*x* 21
**all real num except x **5 2**3; **

**(4, 0), (0, **2**4); removable**

**all real num except **2**1; x **5 2**1, no **
**x-intercept, (0, 2); non-removable**

**vertical asymptotes at *** x*5

**1 and**

*5 2*

**x****1**

**vertical asymptote at *** x*5

**1; hole**

**at**

*5*

**x****0**

**vertical asymptote at *** x*5 2

**2;**

**hole at**

*5*

**x****2**

**vertical asymptote at *** x*5

**4**

* y*5

**0**

*5*

**y****1**

**all real num except 2; x **5 **2;**
**(**2**1, 0) (3, 0), (0, 3 _{2}); non-removable**

**all real num; none; (0, 0)**

**vertical asymptote at **
* x*5 2

**2**

**vertical asymptote at x**5**3; **
**hole at x**5 2**3**

**no vertical asymptotes or **
**holes**

**vertical asymptotes at **
* x*5 2

**4**

_{5}and*5*

**x****3**

* y*5

**2**

*5*

**y****0**

*x*
*y*

*O*
2

⫺2

⫺4 4 6 8

⫺4 ⫺6 4 6

*x*
*y*

*O*
2
⫺2

2

⫺2 ⫺4

⫺6 4 6

⫺4 ⫺6 4 6

*O*
2

⫺2 ⫺2

2
*x*
*y*

*O*

2

2

**Prentice Hall Gold Algebra 2 • **Teaching Resources
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## 8-3

**Practice **

(continued) Form *G*

### Rational Functions and Their Graphs

** 21. ** How many milliliters of 0.75% sugar solution must be added to 100 mL of 1.5%
sugar solution to form a 1.25% sugar solution?

** 22. ** A soccer player has made 3 of his last 24 shots on goal, or 12.5%. How many more
consecutive goals does he need to raise his shots-on-goal average to at least 20%?

** 23. Error Analysis** A student listed the asymptotes of the
function *y* 5 *x*215*x* 16

*x*(*x*2 14*x* 14) as shown at the right. Explain
the student’s error(s). What are the correct asymptotes?

**Sketch the graph of each rational function.**

** 24. ** *y*5 *x*

*x*(*x* 26) **25. ***y* 5
2*x*

*x* 26 **26. ***y*5

*x*221

*x*224 **27. ***y* 5

2*x*2110*x* 112

*x*2 29

** 28. **You start a business word-processing papers for other students. You spend
$3500 on a computer system and offi ce furniture. You fi gure additional costs at
$.02 per page.

** a. **Write a rational function modeling the total average cost per page.
Graph the function.

** b. **What is the total average cost per page if you type 1000 pages? If
you type 2000?

** c. **How many pages must you type to bring your total average cost
to less than $1.50 per page?

** d. **What are the vertical and horizontal asymptotes of the graph of the
function?

horizontal asymptote none

vertical asymptote x = 0

**50 mL**

**3**

**The horizontal asymptote should be *** y*5

**0, because the**

**degree of the numerator is less than the degree of the**

**denominator. The zeros of the denominator are x**5

**0**

**and x**5 2

**2, so there should also be a vertical asymptote**

**at**

*5 2*

**x****2.**

* x*5

**0;**

*5*

**y****0.02**

**$3.52; $1.77**

**at least 2365 pages**

*x*
*y*

*O*
2
⫺2

2 ⫺2 ⫺4

⫺6 4

⫺4 ⫺6 4 6

*x*
*y*

3 ⫺3

⫺6 6 9

⫺6 ⫺9 6

* y*5

**0.02x**1

_{x}**3500, where x**5

**number of pages**

*x*

*y*

*O*
2
⫺2

2

⫺2 4 6 8 10 ⫺4

⫺6 4 6

*x*
*y*

*O*
⫺3

3

⫺3

⫺6 9 12

⫺6

**Prentice Hall Foundations Algebra 2 • **Teaching Resources
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## 8-3

**Practice **

*Form K*

### Rational Functions and Their Graphs

**Find the domain, points of discontinuity, and ****x****- and ****y****-intercepts of each **
**rational function. Determine whether the discontinuities are removable or **
**non-removable. To start, factor the numerator and denominator, if possible.**

** 1. ***y*5 *x _{x}*1

_{2}5

_{2}

**2.**

*y*5 1

*x*212*x*11 **3. ***y*5

*x*14

*x*212*x*28

**Find the vertical asymptotes and holes for the graph of each rational function.**

** 4. ***y*5 *x _{x}*1

_{1}6

_{4}

**5.**

*y*5 (

*x*2

*2)(*

_{x}_{2}

*x*

_{2}21)

**6.**

*y*5

*x*11 (3

*x*22)(

*x*23)

**Find the horizontal asymptote of the graph of each rational function. To start, **
**identify the degree of the numerator and denominator.**

** 7. ***y*5 *x _{x}*1

_{1}1

_{5}

**8.**

*y*5

*x*12

2*x*2 24 **9. ***y*5

3*x*3 24
4*x* 11

*x*11

*x*15

ddegree 1 ddegree 1

**Sketch the graph of each rational function.**

** 10. ** *y*5 *x* 12

(*x* 13)(*x*24) **11. ***y*5

*x* 13

(*x* 21)(*x*25) **12. ***y*5
2*x*

3*x* 21
**domain: all real numbers **

**except x**5**2; non-removable **
**point of discontinuity at **
* x*5

**2; x-intercept:**

*5 2*

**x****5;**

*5 2*

**y-intercept:****y****5**

_{2}**vertical asymptote: *** x*5 2

**4**

* y*5

**1**

**domain: all real numbers **
**except *** x*5 2

**1;**

**non-removable point of **
**discontinuity at *** x*5 2

**1;**

**x-intercept: none;***5*

**y-intercept:****y****1**

**vertical asymptote: none; **
**hole at *** x*5

**2**

* y*5

**0**

**domain: all real numbers **
**except *** x*5

**2 and**

*5 2*

**x****4;**

**non-removable point of**

**discontinuity at**

*5*

**x****2 and**

**removable point of discontinuity**

**at**

*5 2*

**x****4; x-intercept: none;**

*5 2*

**y-intercept:****y****1**

_{2}**vertical asymptotes: *** x*5

**2**

_{3}and*5*

**x****3**

**no horizontal asymptote**

*O*

*x*
*y*

4 8

⫺4 ⫺8

⫺4 ⫺8

4 8 *O* *x*

*y*

4

⫺4 ⫺4 ⫺8

8

4 *O* *x*

*y*

4

⫺4 ⫺8

⫺4 ⫺8

**Prentice Hall Foundations Algebra 2 • **Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

** 13. **Th e CD-ROMs for a computer game can be manufactured for $.25 each. Th e
development cost is $124,000. Th e fi rst 100 discs are samples and will not be sold.
** a. **Write a function for the average cost of a disc that is not a sample.

** b. **What is the average cost if 2000 discs are produced? If 12,800 discs are
produced?

** c. Reasoning** How could you fi nd the number of discs that must be produced
to bring the average cost under $8?

** d. **How many discs must be produced to bring the average cost under $8?

** 14. Error Analysis** For the rational function *y*5 *x*222*x*28

*x*229 , your friend said
that the vertical asymptote is *x* 51 and the horizontal asymptotes are *y* 53
and *y* 5 23. Without doing any calculations, you know this is incorrect.
Explain how you know.

**Sketch the graph of each rational function.**

** 15. ** *y*5 4*x*22100

2*x*2 1*x*215 **16. ***y*5
2*x*2

5*x* 11 **17. ***y*5

2

*x*224

** 18. Multiple Choice** What are the points of discontinuity for the graph of

*y*5 (2*x* 13)(*x* 25)
(*x* 15)(2*x* 21)?

25, 1 23_{2}, 5 25, 1_{2} 5, 21_{2}

## 8-3

**Practice **

(continued) Form *K*

### Rational Functions and Their Graphs

* y*5

**0.25x**

*1*

_{x}_{2}

**124,000**

_{100}**c. The intersection of y**5**0.25x***_{x}*1

_{2}

**124,000**

_{100}**and y**5

**8 rounded to the next whole number**

**gives the number of discs that must be produced.**

**A rational function can have only 1 horizontal asymptote. Your friend must have **
**switched the horizontal and vertical asymptote answers.**

**$65.53; $10.02**

**16,104 discs**

*O* *x*

*y*

4 8

⫺8 ⫺4

*O* *x*

*y*

4 8

⫺4 ⫺8

⫺4 ⫺8

4 8

*O*
*x*
*y*

2

⫺2 ⫺4

⫺2 ⫺4

2

**Prentice Hall Algebra 2 • **Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

## 8-3

**Standardized Test Prep**

_{Rational Functions and Their Graphs}

**Multiple Choice**

**For Exercises 1−4, choose the correct letter.**

** 1. **What function has a graph with a removable discontinuity at Q5, 1_{9}R?

*y* 5 (*x* 25)

(*x*14)(*x* 25) *y* 5

4*x* 21
5*x* 11

*y* 5_{x}_{2}4 _{5} *y* 5_{5}*x _{x}*1

_{2}1

_{4}

** 2. **What is the vertical asymptote of the graph of *y* 5(*x*12)(*x*23)

*x*(*x*23) ?

*x* 5 23 *x* 5 22 *x*5 0 *x*53

** 3. **What best describes the horizontal asymptote(s), if any, of the graph of

*y*5 *x*212*x*28
(*x*16)2 ?

*y* 5 26 *y* 51

*y* 50 Th e graph has no horizontal asymptote.

** 4. **Which rational function has a graph that has vertical asymptotes at *x* 5*a* and

*x*5 2*a*, and a horizontal asymptote at *y* 50?

*y* 5(*x*2*a*)(* _{x}x* 1

*a*)

*y*5

*x*2

*x*2 2*a*2

*y* 5 1

*x*2 2*a*2 *y* 5

*x*2*a*

*x*1*a*

**Short Response**

** 5. **How many milliliters of 0.30% sugar solution must you add to 75 mL of 4% sugar
solution to get a 0.50% sugar solution? Show your work.

**A**

**H**

**C**

**G**

** 75(0.04)**1**0.003x**

**75**1 * x* 5

**0.005**

**3**1**0.003x**5**0.375**1 **0.005x**
* x*5

**1312.5 mL**

**[2] correct answer with work shown**

**Prentice Hall Algebra 2 • **Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

## 8-3

**Enrichment**

_{Rational Functions and Their Graphs}

**Other Asymptotes**

Recall that a rational function does not have a horizontal asymptote if the degree
of the numerator is greater than the degree of the denominator. If, however, the
degree of the numerator is exactly one more than the degree of the denominator,
then the graph of the function has a **slant asymptote**.

You can use long division to fi nd the equation of a slant asymptote. Th e equation of the slant asymptote is given by the quotient, disregarding the remainder.

** 1. **Use long division to fi nd the slant asymptote of *f*(*x*) 5*x _{x}*2

_{1}2

_{1}

*x*.

**Th e slant asymptote of the function is**

*y*5 .

**2.**Graph the slant asymptote on a coordinate grid.

** 3. **Graph the vertical asymptote for the equation in Exercise 1 on the grid.
** 4. **Copy and complete the table, and plot the points accordingly.

Connect with a smooth curve, being sure to draw near to all asymptotes.

** 5. **Use long division to fi nd the slant asymptote of *f*(*x*) 5 *x*3

*x*2 11.
Th e slant asymptote of the function is *y*5 .

** 6. **Graph the slant asymptote on a new coordinate grid.

** 7. **Find any vertical asymptotes for the equation in Exercise 5 and graph on the grid.
** 8. **Copy and complete the table, and plot the points accordingly.

Connect with a smooth curve, being sure to draw near to all asymptotes.

** 9. **Th e technique used to fi nd slant (linear) asymptotes works for rational functions in which
the degree of the numerator is one more than the degree of the denominator. Use long

** ** division to fi nd the non-linear asymptote of the rational function given by *f*(*x*)5*x*3* _{x}*11.
Th e asymptote of the function is

*y*= . Can you guess the

shape of this asymptote?
**x**

**f(x)**

0 2

22

23 3

**x**

**f(x)**

0 2

22

23 3

* x*2

**2**

**x**

**x****2**

4

⫺4

⫺4 4

*x*
*y*

*O*

2

⫺2 ⫺2

2

*x*
*y*

2** _{6}** 2

**6**

**0**

**2**

_{3}**3**

_{2}**0** **8 _{5}**

**8**
**5**

2**27**
**10**

**27**
**10**

2

**Prentice Hall Algebra 2 • **Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

## 8-3

**Reteaching**

_{Rational Functions and Their Graphs}

A rational function may have one or more types of discontinuities: holes (removable points of discontinuity), vertical asymptotes (non-removable points of discontinuity), or a horizontal asymptote.

**If** **Then** **Example**

*a* is a zero with multiplicity

*m *in the numerator and
multiplicity *n* in the
de nominator, and *m*

### $

*n*

hole at *x*5*a*

*f*(*x*)5(*x*25)(*x*16)
(*x* 25)
hole at *x*5 5

*a* is a zero of the
denominator only, or *a*

is a zero with multiplicity

*m* in the numerator and
multi plicity *n* in the
denominator, and *m*

### ,

*n*

vertical asymptote

at *x*5*a* *f*(*x*)5

*x*2
*x* 23

vertical asymptote at *x*5 3

Let *p *5degree o f numerator.
Let *q *5degree of denominator.

• *m*, *n* horizontal asymptote at *y* 50

*f*(*x*)5 4*x*2
7*x*2 12

horizontal asymptote at *y*5 4_{7}
• *m*. *n* no horizontal asymptote exists

• *m*5 *n* _{horizontal asymptote at }_{y}_{5}*a*

*b*,

where *a* and *b* are coeffi cients
of highest degree terms in
numerator and denominator

**Problem**

What are the points of discontinuity of *y* 5*x*2 1*x*26

3*x*2 212 , if any?

**Step 1** Factor the numerator and denominator completely. *y* 5 (*x* 22)(*x* 13)
3(*x* 22)(*x*12)
**Step 2** Look for values that are zeros of both the numerator and the denominator.

Th e function has a hole at *x* 52.

**Step 3** Look for values that are zeros of the denominator only. Th e function has a
vertical asymptote at *x* 5 22.

**Step 4** Compare the degrees of the numerator and denominator. Th ey have the same
degree. Th e function has a horizontal asymptote at *y* 51_{3}.

**Exercises**

**Find the vertical asymptotes, holes, and horizontal asymptote for the graph of **
**each rational function.**

** 1. ***y*5 *x*

*x*229 **2. ***y* 5

6*x*226

*x* 21 **3. ***y*5

4*x* 15
3*x* 12
**horizontal asymptote: **

* y *5

**0**

**vertical asymptotes: x **5** 3, x **52**3;** **hole: x **5** 1**

**Prentice Hall Algebra 2 • **Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

## 8-3

**Reteaching **

(continued)
### Rational Functions and Their Graphs

Before you try to sketch the graph of a rational function, get an idea of its general shape by identifying the graph’s holes, asymptotes, and intercepts.

**Problem**

What is the graph of the rational function *y* 5*x _{x}*1

_{1}3

_{1}?

**Step 1** Identify any holes or asymptotes.

no holes; vertical asymptote at *x*5 21; horizontal asymptote at *y* 51_{1}5 1
**Step 2** Identify any *x*- and *y*-intercepts.

*x*-intercepts occur when *y* 50.*y*-intercepts occur when *x*50.

*x*13

*x*11 50 *y*5

013 011

*x*13 50 *y*5 3

*x*5 23

*x*-intercept at 23 *y*-intercept at 3

**Step 3** Sketch the asymptotes and **Step 4** Make a table of values, plot the points,

intercepts. and sketch the graph.

**Exercises**

**Graph each function. Include the asymptotes. **

** 4. ***y*5 4

*x*229 **5. ***y*5

*x*212*x*22

*x*21
*O*

⫺1

⫺3 ⫺2 2 3

2

1 3

⫺2 ⫺3

*x*

*y* _{x}_{y}

22

21.5

1

3

21

23

2

1.5 _{O}

⫺1

⫺3 ⫺2 2 3

2

1 3

⫺2 ⫺3

*x*
*y*

⫺2 2 4

⫺4

⫺2

⫺3

1 2

*x*
*y*

⫺4 4 6 8

⫺8 _{⫺}

4 4 8

**Prentice Hall Algebra 2 • **Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

## 8-4

**Additional Vocabulary Support **

_{Rational Expressions}

**Th ere are two sets of cards that show how to simplify ** **x****2**2**4**
**x****2** 2**2*** x*1

**1**?

**x****2** 1 **2*** x*2

**3**

**2**

**x****2**2

**3**

*2*

**x****2.**

**Th e set on the left explains the thinking. Th e set on the right shows the steps.**

**Write the steps in the correct order.**

**Think Cards ** **Write Cards**

Factor the numerators and

denominators. (x

22)(x12)

(x 21)(x21) ?

(x 13)(x21)

(2x11)(x22)

(x 12)(x13)

(x21)(2x 11)

x2 _{2}_{4}

x2 22x11?

x2 _{1}_{2x}_{2}_{3}

2x2 23x22

(x22)(x12)

(x 21)(x21) ?

(x13)(x21)

(2x 11)(x22) Write the problem.

Write the remaining factors.

Divide out common factors.

**Step 1 **

**Step 2 **

**Step 3 **

**Step 4 **
**First, **

**Second, **

**Third, **

**Fourth, **

** Think ** **Write**

**write the problem.**

**factor the numerators **
**and denominators.**

**divide out common **
**factors.**

**write the remaining **
**factors.**

**x****2**_{2}_{4}**x****2**_{2}_{2x}_{1}** _{1}**?

**x****2** _{1}_{2x}_{2}_{3}**2x2**_{2}_{3x}_{2}_{2}

**(x**2**2)(x**1**2)**
**(x**2**1)(x**2**1)**?

**(x**1**3)(x**2**1)**
**(2x**1**1)(x**2**2)**

**(x**2**2)(x**1**2)**
**(x**2**1)(x**2**1)** ?

**(x**1**3)(x**2**1)**
**(2x**1**1)(x**2**2)**